232 XIII INTEGRATION (b) For each integer /i>0, let Ai= [2~n, 3 -2-"'1! and A == [3 •2~n~1, 2~n + 1[. Let B; = A^XA;, B; = A;XAJ, C; = A;XA;, C; = A;XA; in R2. Define /:R2->R as follows: f(x,y) = 4n+l for (*, ^) e BJ u BJ; /(x,^)=-4n + 1 for (x, 7) e C£ u C, for each integer « > 0; /(#, X) = 0 otherwise. Show that / is measur- able and that the two integrals \dy \f(x, y) dx and I dx \f(x, y) dy are defined and equal, but that/is not integrable with respect to Lebesgue measure on R2. 4. Let X, Y be two locally compact spaces, A a positive measure on X and p, a positive measure on Y. Let /be a mapping of X x Y into a metrizable space G, such that: (1) for each x e X, the mapping/^,') is ^-measurable; (2) for each y e Y, the mapping /(•,y) is continuous. Show that under these hypotheses /is (A(x)ft)-measurable. (Reduce to the case where X and Y are compact; by using EgorofT's theorem and the fact that X is metrizable, show that / is almost everywhere (with respect to A ® p) the limit of a sequence of (A (x) /^-measurable functions.) 5. Let X, Y be two locally compact spaces, A a positive measure on X and /x a positive measure on Y. Let/be a real-valued function ^0 on X x Y which is bounded on every compact subset of X x Y and such that: (1) for almost all x e X the function /(*, •) is jit-measurable; (2) for each function h e JT(Y), the function XH+J/(x, y)h(y) dp.(y)t which is defined almost everywhere, is A-measurable. Show that under these condi- tions there exists a (A (x) /immeasurable function g such that for each x e X we have f(xt y) — g(x, y) except at the points of a /x-negligible set A* (depending on x). (Show that, for each function ue Jf(X x Y), the function /(x, -)w(x,') is /Lt-integ- rable for almost all xeX, and that the function XH-+ \f(x,y)u(x,y) dp(y\ defined almost everywhere, is A-integrable; for this purpose approximate u by functions of the form v(x)w(y). Then remark that the linear form KI-* dX(x) \f(xty)u(xty)dp,(y) is a positive measure on X x Y with base A ® jit, and apply the Lebesgue-Nikodym theorem. Finally use the fact that Y is a denumerable union of relatively compact open sets Un and that there exists in Jf (Y) a denumerable set of functions D such that every function belonging to JT(Y) is the uniform limit of functions belonging to D and having their supports contained in some Un.) (b) Show that the conditions of (a) are satisfied if: (1) for almost all ^eY, the function /(-,y) is A-measurable; (2) for almost all xeX, the function f(x, •) is continuous almost everywhere with respect to /x. (Use Problem 7(c) of Section 13.9.) (c) Take X = Y = [0,1 ] and take A and ^ to be Lebesgue measure. Assuming the continuum hypothesis, let x < y be a well-ordering on X for which there exists no greatest element and such that, for each x e X, the set of all z e X such that z < x is denumerable. Show that the characteristic function/of the set of pairs (x, y) for which x < y satisfies the conditions of (a), but that f(x, y) d\(x) = 0 for all y E Y and f/(x, y) dptf) - 1 for all x € X. \> 6. Let w, v be two increasing real-valued functions on R, each continuous on the right, and such that u(x) = v(x) = 0 for x < 0. Let w be the increasing function on R, continuous on the right, defined by w(t) — u(t)v(t) for / J> 0 and w(r) = 0 for / < 0. Let A, jLt,, v be the Stieltjes measures associated with w, y, w, respectively (Section 13.18, Problem 6).pology «^~4, v (use (e)). Give an example of a